Rewrite cot^2(x)cos(x)/(csc(x)-1) as an expression that doesn’t include a fraction

We are given the expression:

\[
\frac{\cot^2(x) \cos(x)}{\csc(x) - 1}
\]

### Step 1: Rewrite in terms of sine and cosine
First, express \(\cot(x)\) and \(\csc(x)\) in terms of sine and cosine:

\[
\cot(x) = \frac{\cos(x)}{\sin(x)} \quad \text{and} \quad \csc(x) = \frac{1}{\sin(x)}
\]

Now substitute these into the original expression:

\[
\frac{\left( \frac{\cos^2(x)}{\sin^2(x)} \right) \cos(x)}{\frac{1}{\sin(x)} - 1}
\]

### Step 2: Simplify the numerator and denominator
First, simplify the numerator:

\[
\frac{\cos^2(x)}{\sin^2(x)} \cdot \cos(x) = \frac{\cos^3(x)}{\sin^2(x)}
\]

Now simplify the denominator:

\[
\frac{1}{\sin(x)} - 1 = \frac{1 - \sin(x)}{\sin(x)}
\]

### Step 3: Combine the fraction
The expression now looks like this:

\[
\frac{\frac{\cos^3(x)}{\sin^2(x)}}{\frac{1 - \sin(x)}{\sin(x)}}
\]

This is a complex fraction. To simplify it, multiply the numerator by the reciprocal of the denominator:

\[
\frac{\cos^3(x)}{\sin^2(x)} \cdot \frac{\sin(x)}{1 - \sin(x)} = \frac{\cos^3(x) \cdot \sin(x)}{\sin^2(x) \cdot (1 - \sin(x))}
\]

### Step 4: Simplify the result
Cancel one factor of \(\sin(x)\) from the numerator and denominator:

\[
\frac{\cos^3(x)}{\sin(x) \cdot (1 - \sin(x))}
\]

This is the expression without a fraction.

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Would you like more details on any step or further clarification?

Here are five related questions to expand on this topic:
1. How can we simplify expressions involving both sine and cosine further?
2. How do cotangent and cosecant relate to other trigonometric identities?
3. What strategies can be used to simplify complex fractions?
4. How does this expression behave for specific values of \(x\), such as \(\pi/2\) or \(\pi\)?
5. What are some real-world applications of simplifying trigonometric expressions?

**Tip:** When dealing with trigonometric identities, converting everything into sine and cosine often simplifies the process!

Rewrite cot^2(x)cos(x)/(csc(x)-1) as an expression that doesn’t include a fraction.

This step-by-step solution shows how to simplify the trigonometric expression cot^2(x)cos(x)/(csc(x)-1) into a non-fraction form using trigonometric identities and simplification techniques. Learn key concepts from advanced trigonometry, including cotangent, cosecant, and sine-cosine relationships. Suitable for students in Grades 11-12.

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